Problem 16 - Algebra 2

Question

For each situation, find the margin of error for the sample. Then find an interval likely to contain the true population proportion. Of 400 teenagers surveyed, 62\(\%\) do not plan to stay in their community after finishing their education.

Step-by-Step Solution

Answer

The margin of error for the sample is approximately 0.047, and the interval likely to contain the true population proportion is (0.573, 0.667).

Step 1: Calculate the Sample Proportion

First, we need to find the proportion of the sample. We are told that 62% of the teenagers surveyed do not plan to stay in their community after finishing their education. This percentage is equivalent to a decimal of 0.62. Therefore, the sample proportion (p̂) is 0.62.

Step 2: Calculate the Margin of Error

The margin of error for a proportion is calculated using the formula: MOE = \(Z\sqrt{ \frac{p̂(1-p̂)}{n} }\) where Z is the z-score (for a 95% confidence interval, Z is approximately 1.96), p̂ is the sample proportion, and n is the sample size. In our case, Z = 1.96, p̂ = 0.62, and n = 400. Substituting these values into the formula, we get MOE = 1.96 * \( \sqrt{\frac{0.62(1-0.62)}{400}} = 0.047\).

Step 3: Find the Confidence Interval

A confidence interval likely to contain the true population proportion can be calculated as (p̂ - MOE, p̂ + MOE). Considering the MOE as 0.047 and the sample proportion (p̂) as 0.62, we find the confidence interval as (0.62 - 0.047, 0.62 + 0.047) = (0.573, 0.667), which means we are 95% confident the true population proportion lies within this interval.

Key Concepts

Sample Proportion

In statistics, a sample proportion is a way of representing the fraction of a sample that exhibits a certain characteristic. In this scenario, we have a sample of 400 teenagers. We discovered that 62% do not plan to stay in their community after finishing their education. To express this percentage as a decimal, we convert 62% to 0.62. Thus, the sample proportion, often denoted as \( \hat{p} \), is 0.62.
Understanding sample proportion is crucial because it helps estimate the traits of a larger population. Consider it as a snapshot or a summary of the survey findings. When dealing with sample proportions, always convert percentages to decimals for accuracy and ease of calculation.

Confidence Interval

A confidence interval provides a range that behaves as an estimate for an unknown population parameter. It tells us that we can be confident, up to a certain percentage, that the true population proportion lies within this range. For example, in this exercise, we calculated a confidence interval of (0.573, 0.667) based on the sample proportion.
To compute a confidence interval, we use the sample proportion and the margin of error. The interval is then formed by subtracting and adding the margin of error to the sample proportion:

• Lower limit = \( \hat{p} - \text{MOE} \)
• Upper limit = \( \hat{p} + \text{MOE} \)

This tells us that at the specified confidence level, normally 95%, the true population proportion is expected to be within the calculated interval. A confidence interval doesn't guarantee that the true proportion is within this range, but it does suggest a high level of confidence.

Population Proportion

The population proportion is what statistics aim to determine more precisely. It's the real fraction of the entire population that exhibits the characteristic of interest - in this case, the percentage of all teenagers, not just the sampled ones, who do not plan to stay in their community.
However, since surveying the entire population is often impractical, we use samples to estimate it. When calculated properly, the confidence interval surrounds this population proportion, suggesting where it might lie with a certain degree of confidence. In simple terms, it is our best guess refined by sample data and statistical methods.
This is why finding a reliable sample proportion, margin of error, and confidence interval helps us approach the true population proportion more accurately than merely guessing.

Z-Score

A z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It represents how many standard deviations a data point is from the average. In confidence interval calculations, the z-score corresponds to the desired level of confidence.
For a 95% confidence interval, the commonly used z-score is approximately 1.96. This value comes from the standard normal distribution, where 95% of the data falls within nearly 1.96 standard deviations from the mean.
Using the z-score helps in scaling the standard deviation of the sample, allowing us to calculate the margin of error. Essentially, the z-score translates the level of confidence into a multiplier used in assessing the reliability of the sample's estimate for the entire population.